## Simplified External Ballistics for 21st Century Bullets

This web site will show several ways to calculate the flat fire trajectory of a rifle bullet in the point mass approximation assuming that samples of the down range velocity profile are available. The velocity profile description is a logical and more complete method than the conventional reference drag function, ballistic coefficient, and form factor, which approach still requires muzzle velocity for a complete trajectory. In large measure this site is the equivalent of an electronic notebook on how to do the calculations. This site will also show how to calculate trajectories with finite difference methods, and with explicit velocity profiles.

Given the increasing cost of bullets and transportation, it is useful to get as much information as possible before pulling the trigger. With calculations we can re-adjust scope zero for different distances. We can find the best point blank range or the sighting adjustments for different bullets when we can't buy the ones we really wanted. We can assess the impact of loading errors or bullet weight variation. And overall, we can better understand how to hit the target.

The study of how bullets fly after they leave the barrel of a gun is the subject of external ballistics. External ballistics also includes arrows, cannon balls, BBs and rockets. After all the things man has thrown or blown into the sky, the study is extensive and complicated. Technologies to non-destructively measure the velocity of bullets are only decades old. Over the centuries since Newton studied ballistics, computational techniques evolved that used assumed standard drag functions which in turn implied standard velocity profiles. This "standard function" method allows the use of tabulated data from which the data specific to a particular circumstance was extracted based on a small number of actual measurements. Most books on the topic include pages and pages of these numbers, none of which will be required here.

On the internet today there are many web sites which offer calculators of bullet trajectories. While these programs may provide accurate profiles, the underlying math and the understanding implicit in it are hidden. The many calculators tell you the "how" but not the "why".

For all the complexity of things that might move through the air, the modern rifle bullet is relatively simple. Having velocity measurements available we can explain the trajectory with three ideas, and to an accuracy of roughly 2 MOA as seen by the shooter, with just four numbers. With more calculation, we can get much higher accuracy. Initially we are concerned with two problems in predicting where the bullet will go. First we consider the down range velocity profile, then from this profile we find (second) the drop of the bullet as it travels.

The ideas behind describing trajectories are that 1) The slowing of a bullet is related to its shape, its speed, and the amount of air it has to move out of the way. We can often describe the resulting velocity profile as a parabola in distance requiring three numbers. 2)Once out of the gun the bullet falls due to gravity 3) The falling rate is moderated by aerodynamic effects. A possible way to think about this is that the bullet creates a region of high density air, and the bullet falls through this dense air less rapidly than in a vacuum. For casual shooting we can find the drop by adjusting the raw 1/2 G T^2 distance.

In the section on trajectory parameters we will show a still more elaborate and more accurate description of the path of a bullet.

Rifleist does not know if this analysis is trivial or profound; most likely some of both for most shooters. Whatever, it is the formal intent of this website to finesse the use of drag functions for practical shooting.

This site is about understanding how bullets fly, as proven by figuring out where they will go. This site supports the view that, given velocity profiles, shooters need not be concerned with drag functions or the math behind them. This site assumes that supersonic trajectories can be handled by empirical functions based on measurement and do not require additional external functions.

The site shows a approximate multiplier to correct gravitational drop for observed drag in eq. 4B as (V/Vz)^.3

Perhaps most importantly, this site shows that the velocity profile effectively decouples the X and Y drag equations. Having decoupled the x and y motions, we show an exact reformulation for arbitrary profiles (BD2)

From the decoupled drag equation, we show a drop function based on the exponential velocity profile, and some real data with which to compare.

The "classical external ballistics" section shows how the parabolic velocity profile works and bridges between the velocity profile method and the reference drag function.

Use anything here at your peril; much of it is untested. It is accurate only to the extent the ballistic equations and their handling is accurate. As it stands, 15A and B in the "Classical Ballistics" section do not match as well as they should, so adjustment of Vz of existing profiles is preferred over using published drag and ballistic coefficient.

Substantive comments can be sent to CuZnfinder@analyticalrifleist.com.

Edits to Trajectory Parameters 5-Dec-2013 to eliminate the drag ratio in some cases. Addition of full accuracy finite difference drag equations 20 Dec 2013. A,B estimates from total drag on Dec. 22. As of 17 Jan there is an exact drop formula for the exponential velocity profile, BD5-A Given that the formula is restricted to the high velocity region of rifle bullets, it is not the long awaited holy grail of external ballistics, but it is solid chip off the edge. (Or perhaps a "handle" to the holy grail?) On 21Jan added table of results of BD5-A. Feb 9 added decoupled x,y velocity eqn BD2. 1Mar: changes in Vz variation. On 13 Mar, added BD8, the Y value for the exponential profile. Note on full velocity profile 12 April. Section on Coriolis and other forces updated 3 July 2014. Bolt to ogive for 7 mm Rem mag Dec. 2014.

Note on Temperature Feb 2016.

**Introduction**Given the increasing cost of bullets and transportation, it is useful to get as much information as possible before pulling the trigger. With calculations we can re-adjust scope zero for different distances. We can find the best point blank range or the sighting adjustments for different bullets when we can't buy the ones we really wanted. We can assess the impact of loading errors or bullet weight variation. And overall, we can better understand how to hit the target.

The study of how bullets fly after they leave the barrel of a gun is the subject of external ballistics. External ballistics also includes arrows, cannon balls, BBs and rockets. After all the things man has thrown or blown into the sky, the study is extensive and complicated. Technologies to non-destructively measure the velocity of bullets are only decades old. Over the centuries since Newton studied ballistics, computational techniques evolved that used assumed standard drag functions which in turn implied standard velocity profiles. This "standard function" method allows the use of tabulated data from which the data specific to a particular circumstance was extracted based on a small number of actual measurements. Most books on the topic include pages and pages of these numbers, none of which will be required here.

On the internet today there are many web sites which offer calculators of bullet trajectories. While these programs may provide accurate profiles, the underlying math and the understanding implicit in it are hidden. The many calculators tell you the "how" but not the "why".

**The Basic Ideas**For all the complexity of things that might move through the air, the modern rifle bullet is relatively simple. Having velocity measurements available we can explain the trajectory with three ideas, and to an accuracy of roughly 2 MOA as seen by the shooter, with just four numbers. With more calculation, we can get much higher accuracy. Initially we are concerned with two problems in predicting where the bullet will go. First we consider the down range velocity profile, then from this profile we find (second) the drop of the bullet as it travels.

The ideas behind describing trajectories are that 1) The slowing of a bullet is related to its shape, its speed, and the amount of air it has to move out of the way. We can often describe the resulting velocity profile as a parabola in distance requiring three numbers. 2)Once out of the gun the bullet falls due to gravity 3) The falling rate is moderated by aerodynamic effects. A possible way to think about this is that the bullet creates a region of high density air, and the bullet falls through this dense air less rapidly than in a vacuum. For casual shooting we can find the drop by adjusting the raw 1/2 G T^2 distance.

In the section on trajectory parameters we will show a still more elaborate and more accurate description of the path of a bullet.

Rifleist does not know if this analysis is trivial or profound; most likely some of both for most shooters. Whatever, it is the formal intent of this website to finesse the use of drag functions for practical shooting.

**What is novel (so far as we know) at this site?**This site is about understanding how bullets fly, as proven by figuring out where they will go. This site supports the view that, given velocity profiles, shooters need not be concerned with drag functions or the math behind them. This site assumes that supersonic trajectories can be handled by empirical functions based on measurement and do not require additional external functions.

The site shows a approximate multiplier to correct gravitational drop for observed drag in eq. 4B as (V/Vz)^.3

Perhaps most importantly, this site shows that the velocity profile effectively decouples the X and Y drag equations. Having decoupled the x and y motions, we show an exact reformulation for arbitrary profiles (BD2)

From the decoupled drag equation, we show a drop function based on the exponential velocity profile, and some real data with which to compare.

The "classical external ballistics" section shows how the parabolic velocity profile works and bridges between the velocity profile method and the reference drag function.

**Caveat**Use anything here at your peril; much of it is untested. It is accurate only to the extent the ballistic equations and their handling is accurate. As it stands, 15A and B in the "Classical Ballistics" section do not match as well as they should, so adjustment of Vz of existing profiles is preferred over using published drag and ballistic coefficient.

**Where to Complain**Substantive comments can be sent to CuZnfinder@analyticalrifleist.com.

**What Was Last Added**Edits to Trajectory Parameters 5-Dec-2013 to eliminate the drag ratio in some cases. Addition of full accuracy finite difference drag equations 20 Dec 2013. A,B estimates from total drag on Dec. 22. As of 17 Jan there is an exact drop formula for the exponential velocity profile, BD5-A Given that the formula is restricted to the high velocity region of rifle bullets, it is not the long awaited holy grail of external ballistics, but it is solid chip off the edge. (Or perhaps a "handle" to the holy grail?) On 21Jan added table of results of BD5-A. Feb 9 added decoupled x,y velocity eqn BD2. 1Mar: changes in Vz variation. On 13 Mar, added BD8, the Y value for the exponential profile. Note on full velocity profile 12 April. Section on Coriolis and other forces updated 3 July 2014. Bolt to ogive for 7 mm Rem mag Dec. 2014.

Note on Temperature Feb 2016.